[1]} consider SI and SIR epidemics with rewiring on graphs generated by the configuration model (see, for example, [2]}, Chapter 3, and [3]}, Chapter 7) or its close relatives, which include Erdős-Rényi random graphs. Briefly, the configuration model random graph on \(n\)  individuals, labelled \(1,2,\dots , n\) , is constructed as follows. Let \(D_1, D_2, \dots , D_n\)  be i.i.d. copies of a random variable \(D\) , which takes values in \(\mathbb {Z}_+\) . Attach \(D_i\)  half-edges to individual \(i\)  (\(i=1,2,\dots ,n\) ) and then pair up the \(D_1+D_2+\dots +D_n\)  half-edges uniformly at random to form the graph. (If \(D_1+D_2+\dots +D_n\)  is odd then there is a left-over half-edge, which is ignored.) This yields the version of the model introduced in [4]}. In an alternative version ([5]}) the degrees are prescribed deterministically. The configuration model was first introduced by [6]}.
